Spectral convergence in geometric quantization on $K3$ surfaces

We study the geometric quantization on $K3$ surfaces from the viewpoint of the spectral convergence. We take a special Lagrangian fibrations on the $K3$ surfaces and a family of hyper-Kähler structures tending to large complex structure limit, and show a spectral convergence of the $\overline{\partial}$ Laplacians on the prequantum line bundle to the spectral structure related to the set of Bohr–Sommerfeld fibers.

Keywords

geometric quantization, $K3$ surface, Bohr–Sommerfeld fiber, measured Gromov–Hausdorff convergence

2010 Mathematics Subject Classification

53D50, 58C40

Full Text (PDF format)

Received 23 May 2022

Accepted 8 March 2023

Published 7 November 2023